Quintic NLW/NLKG on R

• Scaling is $s_{c}=0$.
• LWP for $s\geq 3/10$ by energy estimates and Sobolev (solution is in $L_{x}^{5}$)
  • For $s<3/10$ one has ill-posedness CtCoTa-p2, indeed it is not even possible to make sense of solutions in the distributional sense.
• GWP for $s\geq 1$ in the defocussing case from energy conservation.
  • It is overwhelmingly likely that one can lower this s index.
  • In the focussing case there is blowup from large data by the ODE method.